[Lemma 6.11 + Claim A.1 [1]}] 
Let \(G\)  be a graph \(G = (V,E)\) . Let \((\tilde{A},\tilde{B})\)  be a distribution over vertex pairs generated as follows. Sample \(A^{\prime } \sim V^R\)  and let \(\tilde{A} \sim T^V_{1 - \eta ^{\prime }}(A^{\prime })\)  and let \(\tilde{B} \sim G^{\otimes R}_{1 - \eta ^{\prime }}(A^{\prime })\) . Let \(F:V^R \rightarrow [R]\)  be an assignment satisfying the following.
\( \Pr _{\tilde{A},\tilde{B}} \Pr _{\pi _{\tilde{A}},\pi _{\tilde{B}} \sim {S}_R} \left[\pi ^{-1}_{\tilde{A}}\left(F\left(\pi _{\tilde{A}} \circ \tilde{A}\right)\right) = \pi ^{-1}_{\tilde{B}}\left(F\left(\pi _{\tilde{B}} \circ \tilde{B}\right)\right) \right] \ge \zeta .\) 
